What a complex number is and why arithmetic on it works
A complex number is a pair of real numbers written a + bi, where i is a symbol obeying one rule: i² = −1. The a is the real part and the b is the imaginary part, and both are ordinary real numbers — the imaginary part of 3 + 4i is 4, not 4i. Every rule of algebra you already know carries over unchanged, because complex numbers form a field: you can add, subtract, multiply and divide freely, and the results obey commutativity, associativity and distributivity exactly as real numbers do.
Complex numbers exist because polynomials demand them. x² + 1 = 0 has no real solution, and once you admit one root the fundamental theorem of algebra says every polynomial of degree n has exactly n roots counted with multiplicity. That is why the quadratic formula stops failing when the discriminant is negative, and why the eigenvalues of a rotation matrix are complex even though every entry of the matrix is real.
The picture that makes the arithmetic intuitive is the Argand diagram: plot the real part along the horizontal axis and the imaginary part along the vertical one, and each complex number becomes a point, or an arrow from the origin. Addition is then vector addition — the same head-to-tail rule as the vector calculators. Multiplication is the interesting one, because it is not vector-like at all: it scales the length and rotates the angle.
The four operations, and why division needs the conjugate
Addition and subtraction act on the two parts independently: (a + bi) ± (c + di) = (a ± c) + (b ± d)i. Nothing mixes, because you can only collect like terms.
Multiplication is binomial expansion followed by one substitution. (a + bi)(c + di) = ac + adi + bci + bdi², and replacing i² with −1 gives (ac − bd) + (ad + bc)i. The minus sign in the real part is the whole content of i² = −1, and it is where nearly every hand-calculation slip happens.
Division starts from a problem: you cannot leave i in a denominator and read off the two parts. The fix uses the conjugate, written z̄ — the same number with the sign of the imaginary part flipped. A number times its conjugate is always real, because (c + di)(c − di) = c² − (di)² = c² + d². Multiply the top and bottom of the fraction by z̄2 and the denominator becomes that real number, leaving (ac + bd) + (bc − ad)i over c² + d².
Polar form makes multiplication and division trivial. Write z = r(cos θ + i sin θ), where r = √(a² + b²) is the modulus and θ = atan2(b, a) is the argument. Then multiplying two complex numbers multiplies their moduli and adds their arguments; dividing divides the moduli and subtracts the arguments. Euler's identity eiθ = cos θ + i sin θ turns that into ordinary index laws: r1eiθ1 × r2eiθ2 = r1r2ei(θ1+θ2).
Worked example: (3 + 4i) ÷ (1 − 2i), checked two ways
Set z1 = 3 + 4i, z2 = 1 − 2i and choose divide.
- Conjugate of the denominator. z̄2 = 1 + 2i.
- New denominator. (1 − 2i)(1 + 2i) = 1² + 2² = 1 + 4 = 5.
- New numerator. (3 + 4i)(1 + 2i) = 3 + 6i + 4i + 8i² = 3 − 8 + 10i = −5 + 10i.
- Divide through. (−5 + 10i) ÷ 5 = −1 + 2i.
- Modulus. √((−1)² + 2²) = √5 = 2.2360680.
- Argument. The point (−1, 2) is in the second quadrant. The reference angle is atan(2 ÷ 1) = 63.434949°, so θ = 180° − 63.434949° = 116.565051°, which is 2.0344439 radians.
- Exponential form. 2.2360680 ei × 2.0344439.
Now check it in polar form, which uses none of the same arithmetic. |z1| = √(9 + 16) = 5 and arg z1 = atan(4 ÷ 3) = 53.130102°. |z2| = √(1 + 4) = 2.2360680 and arg z2 = −63.434949°. Dividing: the modulus is 5 ÷ 2.2360680 = 2.2360680, and the argument is 53.130102° − (−63.434949°) = 116.565051°. Both routes agree, which is the check worth doing on any complex division you do by hand.
How to read the modulus and the argument
The modulus is a size and the argument is a direction. Together they say the same thing as the real and imaginary parts, and which pair you want depends on the question. Rectangular form answers “how far right and how far up”; polar form answers “how long and which way round”. Electrical engineers live in polar form because a sinusoidal voltage is naturally described by an amplitude and a phase, and impedance combines by exactly the complex multiplication rule above.
Read the argument carefully, because its range is a convention. This calculator returns the principal argument in (−180°, 180°], which is what atan2 gives and what most software uses. Adding any whole number of full turns describes the same point, so −90° and 270° are the same direction. Two consequences catch people out: the argument on the negative real axis is reported as +180° rather than −180°, and adding two arguments in polar multiplication can produce a value outside the principal range that needs 360° added or subtracted to bring it back.
Zero is the one number with no argument. Its modulus is 0, and every direction from the origin is as good as any other, so the argument output is left blank rather than reported as 0°. That is not a limitation of this page — arg(0) is genuinely undefined.
A useful sanity check on any product or quotient: the modulus of a product must equal the product of the moduli, and the modulus of a quotient the quotient of the moduli. For the worked example above, 5 × 2.2360680 = 11.180340, and |11 − 2i| = √125 = 11.180340. If those two disagree, the error is in the rectangular arithmetic.
Rectangular and polar forms of common complex numbers
| z | Modulus |z| | Argument (degrees) | Argument (radians) | Quadrant |
|---|---|---|---|---|
| 1 | 1.000000 | 0.0000 | 0.000000 | positive real axis |
| i | 1.000000 | 90.0000 | 1.570796 | positive imaginary axis |
| −1 | 1.000000 | 180.0000 | 3.141593 | negative real axis |
| −i | 1.000000 | −90.0000 | −1.570796 | negative imaginary axis |
| 1 + i | 1.414214 | 45.0000 | 0.785398 | first |
| −1 + i | 1.414214 | 135.0000 | 2.356194 | second |
| −1 − i | 1.414214 | −135.0000 | −2.356194 | third |
| 1 − i | 1.414214 | −45.0000 | −0.785398 | fourth |
| √3 + i | 2.000000 | 30.0000 | 0.523599 | first |
| 3 + 4i | 5.000000 | 53.1301 | 0.927295 | first |
| −3 + 4i | 5.000000 | 126.8699 | 2.214297 | second |
| 5 − 12i | 13.000000 | −67.3801 | −1.176005 | fourth |
The three 5-modulus and 13-modulus entries come from the 3-4-5 and 5-12-13 Pythagorean triples, which is why their moduli are whole numbers. Note that −3 + 4i and 3 + 4i share a modulus but differ in argument by 180° − 2 × 53.1301° = 73.7398°.
The powers of i repeat every four steps
i1 = i, i2 = −1, i3 = −i, i4 = 1, and then the cycle restarts. So to evaluate in for any whole n, take n modulo 4 and look up the answer: i27 has 27 mod 4 = 3, so i27 = −i. In polar terms this is obvious — i has modulus 1 and argument 90°, so raising it to the nth power leaves the modulus at 1 and turns the argument to 90n degrees, which returns to where it started after four quarter-turns. The modulo calculator handles the index reduction for large exponents.
Mistakes and conventions worth checking
- Dropping the minus sign in ac − bd. The real part of a product subtracts. If your answer has the wrong sign on the real part, this is almost always why.
- Entering 4i instead of 4. The imaginary-part field takes the coefficient only. For 3 − 5i, enter 3 and −5.
- Using atan(b/a) instead of atan2(b, a). Plain arctangent cannot distinguish the first quadrant from the third, or the second from the fourth, because dividing b by a throws away both signs. It puts −1 + 2i at −63.43° instead of 116.57°.
- Assuming −180° and +180° are interchangeable in output. They describe the same direction, but the principal-value convention picks +180°, and a program comparing arguments for equality will see them as different.
- Expecting an order relation. There is no consistent way to say one complex number is greater than another. You can compare moduli, and this calculator gives you both, but 3 + 4i is neither larger nor smaller than 5i even though both have modulus 5.
- Reading √(−4) as unambiguous. Both 2i and −2i square to −4. The principal square root convention picks 2i, but any formula that relies on √(zw) = √z·√w for complex arguments is unsafe.
Where complex arithmetic is actually used
Electrical engineering is the largest consumer. A steady sinusoidal voltage or current is represented as a phasor — a complex number whose modulus is the amplitude and whose argument is the phase — and the impedance of an inductor is jωL while a capacitor's is 1 ÷ (jωC). Engineers write j rather than i to avoid a clash with current, but the algebra is identical. Series and parallel combinations then reduce to exactly the multiplication and division on this page, which is why dividing by the conjugate is a working technique rather than a classroom exercise.
Control theory and signal processing use the same objects differently: the poles and zeros of a transfer function are complex numbers, and whether their real parts are negative decides whether a system is stable. The discrete Fourier transform is a sum of terms e−2πikn/N, each one a unit-modulus complex number, and the fast Fourier transform is an exploitation of the fact that those arguments repeat.
Pure mathematics uses complex numbers to make results uniform. Every degree-n polynomial has exactly n complex roots, so a cubic always has three and the awkward case-splitting for real coefficients disappears. Matrices with real entries can have complex eigenvalues, and when they do the pair is always a conjugate pair, which corresponds to a rotation in the plane. If you need vector geometry in three dimensions rather than two, the cross product is the tool — complex numbers only encode rotation in the plane, and the three-dimensional generalisation is quaternions rather than anything on this page.
Key terms
- Imaginary unit i
- The number defined by i² = −1. Engineers usually write j instead, to keep i for current.
- Conjugate
- z̄ = a − bi for z = a + bi. Multiplying a number by its conjugate always gives the real number a² + b², which is the modulus squared.
- Modulus
- |z| = √(a² + b²), the distance from the origin on the Argand plane. Also called the absolute value or magnitude.
- Argument
- arg z = atan2(b, a), the angle from the positive real axis. The principal value is taken in (−180°, 180°].
- Argand diagram
- The plane with the real part on the horizontal axis and the imaginary part on the vertical, on which every complex number is a point.
- Phasor
- A complex number standing for a sinusoid of known frequency, with the modulus giving amplitude and the argument giving phase.